gpcm  R Documentation 
Fits the Generalized Partial Credit model for ordinal polytomous data, under the Item Response Theory approach.
gpcm(data, constraint = c("gpcm", "1PL", "rasch"), IRT.param = TRUE, start.val = NULL, na.action = NULL, control = list())
data 
a 
constraint 
a character string specifying which version of the Generalized Partial Credit Model to fit. See Details and Examples for more info. 
IRT.param 
logical; if 
start.val 
a list of starting values or the character string 
na.action 
the 
control 
a named list of control values with components,

The Generalized Partial Credit Model is an IRT model, that can handle ordinal manifest variables. This model was discussed by Masters (1982) and it was extended by Muraki (1992).
The model is defined as follows
P_{ik}(z) = \frac{\exp ∑ \limits_{c = 0}^k β_i (z  β_{ic}^*)}{ ∑ \limits_{r = 0}^{m_i} \exp ∑ \limits_{c = 0}^r β_i (z  β_{ic}^*)},
where P_{ik}(z) denotes the probability of responding in category k for item i, given the latent ability z, β_{ic}^* are the itemcategory parameters, β_i is the discrimination parameter, m_i is the number of categories for item i, and
∑ \limits_{c = 0}^0 β_i (z  β_{ic}^*) \equiv 0.
If constraint = "rasch"
, then the discrimination parameter β_i is assumed equal for all items and fixed at one. If
constraint = "1PL"
, then the discrimination parameter β_i is assumed equal for all items but is estimated.
If constraint = "gpcm"
, then each item has its one discrimination parameter β_i that is estimated. See
Examples for more info.
If IRT.param = FALSE
, then the linear predictor is of the form β_i z + β_{ic}.
The fit of the model is based on approximate marginal Maximum Likelihood, using the GaussHermite quadrature rule for the approximation of the required integrals.
An object of class gpcm
with components,
coefficients 
a named list with components the parameter values at convergence for each item. 
log.Lik 
the loglikelihood value at convergence. 
convergence 
the convergence identifier returned by 
hessian 
the approximate Hessian matrix at convergence. 
counts 
the number of function and gradient evaluations used by the quasiNewton algorithm. 
patterns 
a list with two components: (i) 
GH 
a list with two components used in the GaussHermite rule: (i) 
max.sc 
the maximum absolute value of the score vector at convergence. 
constraint 
the value of the 
IRT.param 
the value of the 
X 
a copy of the response data matrix. 
control 
the values used in the 
na.action 
the value of the 
call 
the matched call. 
In case the Hessian matrix at convergence is not positive definite try to refit the model by specifying the starting values or using
start.val = "random"
.
gpcm()
can also handle binary items and can be used instead of rasch
and ltm
though it is less
efficient. However, gpcm()
can handle a mix of dichotomous and polytomous items that neither rasch
nor
ltm
can.
Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl
Masters, G. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.
Muraki, E. (1992). A generalized partial credit model: application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.
coef.gpcm
,
fitted.gpcm
,
summary.gpcm
,
anova.gpcm
,
plot.gpcm
,
vcov.gpcm
,
GoF.gpcm
,
margins
,
factor.scores
## The Generalized Partial Credit Model for the Science data: gpcm(Science[c(1,3,4,7)]) ## The Generalized Partial Credit Model for the Science data, ## assuming equal discrimination parameters across items: gpcm(Science[c(1,3,4,7)], constraint = "1PL") ## The Generalized Partial Credit Model for the Science data, ## assuming equal discrimination parameters across items ## fixed at 1: gpcm(Science[c(1,3,4,7)], constraint = "rasch") ## more examples can be found at: ## http://wiki.rproject.org/rwiki/doku.php?id=packages:cran:ltm#sample_analyses
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